Question

For \(g\isin\Z\), let \(\bar{g}\isin\Z_{37}\) denote the residue class of g modulo 37. Consider the group U₃₇ = {\(\bar{g}\isin\Z_{37}:1\le g\le37\) with gcd(g, 37) = 1} with respect to multiplication modulo 37. Then which one of the following is FALSE?

• A. The set \({\bar g \isin U_{37}: \bar g = (\bar g)^{-1}}\) contains exactly 2 elements.
• B. The order of the element \(\overline{10}\) in U₃₇ is 36.
• C. There is exactly one group homomorphism from U₃₇ to (\(\Z\), +).
• D. There is exactly one group homomorphism from U₃₇ to (\(\mathbb{Q}\),+).

Answer 1

The Correct Option is B

To solve this problem, we need to analyze the group \( U_{37} \), which is the set of integers from \( 1 \) to \( 37 \) that are coprime to \( 37 \), under multiplication modulo \( 37 \). This is a standard group theory problem involving modular arithmetic.

The order of an element \(\bar{g}\) in a group is the smallest positive integer \( n \) such that \((\bar{g})^n \equiv 1 \pmod{37}\).

1. First, we determine the elements of \( U_{37} \). Since \( 37 \) is a prime number, the elements of \( U_{37} \) are all integers from \( 1 \) to \( 36 \), because all these integers are coprime to \( 37 \).
2. The group \( U_{37} \) has \( \varphi(37) = 36 \) elements, where \(\varphi\) is the Euler's Totient function.
3. The possible orders of elements in \( U_{37} \) must divide the order of the group, which is \( 36 \). Hence, the possible orders are \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \).
4. Now, consider the statement: "The order of the element \( \overline{10} \) in \( U_{37} \) is 36." To check this, we compute powers of \( 10 \) modulo \( 37 \) to find the smallest positive integer \( n \) such that \( 10^n \equiv 1 \pmod{37} \).
5. Let's compute small powers of \( 10 \) modulo \( 37 \):
   • \( 10^1 \equiv 10 \pmod{37} \)
   • \( 10^2 \equiv 100 \equiv 26 \pmod{37} \)
   • \( 10^3 \equiv 260 \equiv 1 \pmod{37} \)
6. Thus, the statement "The order of the element \( \overline{10} \) in \( U_{37} \) is 36." is indeed FALSE.

The other options would require separate validation, but based on the problem's context, the identified false statement aligns with our findings.